Let $\ell_1$ and $\ell_2$ be the lines $\vec{r}_1 = \lambda(\hat{i} + \hat{j} + \hat{k})$ and $\vec{r}_2 = (\hat{j} - \hat{k}) + \mu(\hat{i} + \hat{k})$, respectively. Let X be the set of all the planes H that contain the line $\ell_1$. For a plane H, let d(H) denote the smallest possible distance between the points of $\ell_2$ and H. Let $H_0$ be a plane in X for which d($H_0$) is the maximum value of d(H) as H varies over all planes in X. Match each entry in List-I to the correct entries in List-II. | List-I | List-II | | --- | --- | | (P) The value of $d(H_0)$ is | (1) $\sqrt{3}$ | | (Q) The distance of the point $(0, 1, 2)$ from $H_0$ is | (2) $\frac{1}{\sqrt{3}}$ | | (R) The distance of origin from $H_0$ is | (3) $0$ | | (S) The distance of origin from the point of intersection of planes $y = z$, $x = 1$ and $H_0$ is | (4) $\sqrt{2}$ | | | (5) $\frac{1}{\sqrt{2}}$ |

Let ell₁ and ell₂ be the lines r⃗₁ = λ(î + ĵ + k̂) and r⃗₂ = (ĵ - k̂) + μ(î + k̂), respectively. Let X be the set of all the planes H that contain the line ell₁. For a plane H, let d(H) denote the smallest possible distance between the points of ell₂ and H. Let H₀ be a plane in X for which d(H₀) is the maximum value of d(H) as H varies over all planes in X. Match each entry in List-I to the correct entries in List-II. | List-I | List-I $\ell_1: \vec{r}_1 = \lambda(\hat{i} + \hat{j} + \hat{k})$ $\ell_2: \vec{r}_2 = (\hat{j} - \hat{k}) + \mu(\hat{i} + \hat{k})$ $X = \{H : \ell_1 \subset H\}, \quad d(H) = \min_{\text{pt} \in \ell_2} \text{dist}(\text{pt}, H)$ $H_0 \in X: \ d(H_0) = \max_{H \in X} d(H)$ $\text{To do: match List-I with List-II}$ $\ell_1: \vec{r}_1 = \lambda(\hat{i}+\hat{j}+\hat{k}), \quad \vec{a}=(1,1,1)$ $\ell_2: \vec{r}_2 = (0,1,-1) + \mu(1,0,1), \quad \vec{b}=(1,0,1)$ $H \in X: \vec{n}\cdot\vec{r}=0, \quad \vec{n}\cdot(1,1,1)=0$ $\Rightarrow n_1 + n_2 + n_3 = 0$ $\text{Plan: } 1.\ d(H)=\text{dist}(\ell_2,H)$ $2.\ \max d(H) \Rightarrow \vec{n} \Rightarrow H_0$ $3.\ \text{the four distances } (P),(Q),(R),(S)$ $d(H) = 0 \ \text{ if } \ \vec{b}\cdot\vec{n} \neq 0$ $d(H) = \dfrac{|\vec{n}\cdot\vec{B}|}{|\vec{n}|} \ \text{ if } \ \vec{b}\cdot\vec{n} = 0$ $\vec{b}\cdot\vec{n}=0 \Rightarrow n_1 + n_3 = 0$ $n_1+n_2+n_3=0 \Rightarrow n_2=0,\ n_3=-n_1$ $\vec{n} \propto (1,0,-1)$ $H_0: x - z = 0 \Rightarrow x = z$ $d(H_0) = \dfrac{|(1,0,-1)\cdot(0,1,-1)|}{\sqrt{2}} = \dfrac{1}{\sqrt{2}}$ $(P) \to (5)$ $\text{dist}\big((0,1,2),\ x-z=0\big) = \dfrac{|0-2|}{\sqrt{2}} = \sqrt{2}$ $(Q) \to (4)$ $\text{dist}(O, H_0) = 0 \quad (O \in H_0)$ $(R) \to (3)$ $y=z,\ x=1,\ x=z \Rightarrow (1,1,1)$ $\text{dist}\big(O,(1,1,1)\big) = \sqrt{3}$ $(S) \to (1)$ $(P)\to(5),\ (Q)\to(4),\ (R)\to(3),\ (S)\to(1)$ $\Rightarrow \text{option } (2)$